A Jamming-Resistant MAC Protocol forMulti-Hop Wireless NetworksA preliminary version of this article appeared at the 24th International Symposium on Distributed Computing (DISC), 2010.

# A Jamming-Resistant MAC Protocol for Multi-Hop Wireless Networks††thanks: A preliminary version of this article appeared at the 24th International Symposium on Distributed Computing (DISC), 2010.

Andrea Richa    Christian Scheideler    Stefan Schmid    Jin Zhang
Computer Science and Engineering
SCIDSE    Arizona State University
Tempe
AZ 85287    USA; {aricha    jzhang82}@asu.edu
Department of Computer Science
Deutsche Telekom Laboratories
TU Berlin    D-10587 Berlin    Germany
stefan@net.t-labs.tu-berlin.de
###### Abstract

This paper presents a simple local medium access control protocol, called Jade, for multi-hop wireless networks with a single channel that is provably robust against adaptive adversarial jamming. The wireless network is modeled as a unit disk graph on a set of nodes distributed arbitrarily in the plane. In addition to these nodes, there are adversarial jammers that know the protocol and its entire history and that are allowed to jam the wireless channel at any node for an arbitrary -fraction of the time steps, where is an arbitrary constant. We assume that the nodes cannot distinguish between jammed transmissions and collisions of regular messages. Nevertheless, we show that Jade achieves an asymptotically optimal throughput if there is a sufficiently dense distribution of nodes.

## 1 Introduction

The problem of coordinating the access to a shared medium is a central challenge in wireless networks. In order to solve this problem, a proper medium access control (MAC) protocol is needed. Ideally, such a protocol should not only be able to use the wireless medium as effectively as possible, but it should also be robust against attacks. Unfortunately, most of the MAC protocols today can be easily attacked. A particularly critical class of attacks are jamming attacks (i.e., denial-of-service attacks on the broadcast medium). Jamming attacks are typically easy to implement as the attacker does not need any special hardware. Attacks of this kind usually aim at the physical layer and are realized by means of a high transmission power signal that corrupts a communication link or an area, but they may also occur at the MAC layer, where an adversary may either corrupt control packets or reserve the channel for the maximum allowable number of slots so that other nodes experience low throughput by not being able to access the channel. In this paper we focus on jamming attacks at the physical layer, that is, the interference caused by the jammer will not allow the nodes to receive messages. The fundamental question that we are investigating is: Is there a MAC protocol such that for any physical-layer jamming strategy, the protocol will still be able to achieve an asymptotically optimal throughput for the non-jammed time steps? Such a protocol would force the jammer to jam all the time in order to prevent any successful message transmissions. Finding such a MAC protocol is not a trivial problem. In fact, the widely used IEEE 802.11 MAC protocol already fails to deliver any messages for very simple oblivious jammers that jam only a small fraction of the time steps [3]. On the positive side, Awerbuch et al. [2] have demonstrated that there are MAC protocols which are provably robust against even massive adaptive jamming, but their results only hold for single-hop wireless networks with a single jammer, that is, all nodes experience the same jamming sequence.

In this paper, we significantly extend the results in [2]. We present a MAC protocol called Jade (a short form of “jamming defense”) that can achieve a constant fraction of the best possible throughput for a large class of jamming strategies in a large class of multi-hop networks where transmissions and interference can be modeled using unit-disk graphs. These jamming strategies include jamming patterns that can be completely different from node to node. It turns out that while Jade differs only slightly from the MAC protocol of [2], the proof techniques needed for the multi-hop setting significantly differ from the techniques in [2].

### 1.1 Model

We consider the problem of designing a robust MAC protocol for multi-hop wireless networks with a single wireless channel. The wireless network is modeled as a unit disk graph (UDG) where represents a set of honest and reliable nodes and two nodes are within each other’s transmission range, i.e., , if and only if their (normalized) distance is at most 1. We assume that time proceeds in synchronous time steps called rounds. In each round, a node may either transmit a message or sense the channel, but it cannot do both. A node which is sensing the channel may either sense an idle channel (if no other node in its transmission range is transmitting at that round and its channel is not jammed), sense a busy channel (if two or more nodes in its transmission range transmit at that round or its channel is jammed), or receive a packet (if exactly one node in its transmission range transmits at that round and its channel is not jammed).

In addition to these nodes there is an adversary (who may control any number of jamming devices). We allow the adversary to know the protocol and its entire history and to use this knowledge in order to jam the wireless channel at will at any round (i.e, the adversary is adaptive). However, like in [2], the adversary has to make a jamming decision before it knows the actions of the nodes at the current round. The adversary can jam the nodes individually at will, as long as for every node , at most a -fraction of its rounds is jammed, where can be an arbitrarily small constant. That is, has the chance to receive a message in at least an -fraction of the rounds. More formally, an adversary is called -bounded for some and , if for any time window of size and at any node , the adversary can jam at most of the rounds at .

Given a node and a time interval , we define as the number of time steps in that are non-jammed at and as the number of time steps in in which successfully receives a message. A MAC protocol is called -competitive against some -bounded adversary if, for any time interval with for a sufficiently large (that may depend on and ), In other words, a -competitive MAC protocol can achieve at least a -fraction of the best possible throughput.

Our goal is to design a symmetric local-control MAC protocol (i.e., there is no central authority controlling the nodes, and all the nodes are executing the same protocol) that has a constant-competitive throughput against any -bounded adversary in any multi-hop network that can be modeled as a UDG. In order to obtain a more refined picture of the competitiveness of our protocol, we will also investigate so-called -uniform adversaries. An adversary is -uniform if the node set can be partitioned into subsets so that the jamming sequence is the same within each subset. In other words, we require that at all times, the nodes in a subset are either all jammed or all non-jammed. Thus, a 1-uniform jammer jams either everybody or nobody in a round whereas an -uniform jammer can jam the nodes individually at will.

In this paper, we will say that a claim holds with high probability (w.h.p.) iff it holds with probability at least for any constant ; it holds with moderate probability (w.m.p.) iff it holds with probability at least for any constant .

### 1.2 Related Work

Due to the topic’s importance, wireless network jamming has been extensively studied in the applied research fields [1, 5, 6, 22, 26, 27, 28, 30, 31, 37, 38, 39, 40], both from the attacker’s perspective [6, 26, 27, 40] as well as from the defender’s perspective [1, 5, 6, 27, 28, 30, 38, 40]—also in multi-hop settings (e.g. [21, 32, 42, 43, 44]).

Traditionally, jamming defense mechanisms operate on the physical layer [28, 30, 36]. Mechanisms have been designed to avoid jamming as well as detect jamming. Spread spectrum technology has been shown to be very effective to avoid jamming as with widely spread signals, it becomes harder to detect the start of a packet quickly enough in order to jam it. Unfortunately, protocols such as IEEE 802.11b use relatively narrow spreading [20], and some other IEEE 802.11 variants spread signals by even smaller factors [5]. Therefore, a jammer that simultaneously blocks a small number of frequencies renders spread spectrum techniques useless in this case. As jamming strategies can come in many different flavors, detecting jamming activities by simple methods based on signal strength, carrier sensing, or packet delivery ratios has turned out to be quite difficult [27].

Recent work has also studied MAC layer strategies against jamming, including coding strategies [6], channel surfing and spatial retreat [1, 41], or mechanisms to hide messages from a jammer, evade its search, and reduce the impact of corrupted messages [38]. Unfortunately, these methods do not help against an adaptive jammer with full information about the history of the protocol, like the one considered in our work.

In the theory community, work on MAC protocols has mostly focused on efficiency. Many of these protocols are random backoff or tournament-based protocols [4, 7, 17, 18, 25, 34] that do not take jamming activity into account and, in fact, are not robust against it (see [2] for more details). The same also holds for many MAC protocols that have been designed in the context of broadcasting [8] and clustering [24]. Also some work on jamming is known (e.g., [9] for a short overview). There are two basic approaches in the literature. The first assumes randomly corrupted messages (e.g. [33]), which is much easier to handle than adaptive adversarial jamming [3]. The second line of work either bounds the number of messages that the adversary can transmit or disrupt with a limited energy budget (e.g. [16, 23]) or bounds the number of channels the adversary can jam (e.g. [10, 11, 12, 13, 14, 15, 29]).

The protocols in [16, 23] can tackle adversarial jamming at both the MAC and network layers, where the adversary may not only be jamming the channel but also introducing malicious (fake) messages (possibly with address spoofing). However, they depend on the fact that the adversarial jamming budget is finite, so it is not clear whether the protocols would work under heavy continuous jamming. (The result in [16] seems to imply that a jamming rate of is the limit whereas the handshaking mechanisms in [23] seem to require an even lower jamming rate.)

In the multi-channel version of the problem introduced in the theory community by Dolev [13] and also studied in [10, 11, 12, 13, 14, 15, 29], a node can only access one channel at a time, which results in protocols with a fairly large runtime (which can be exponential for deterministic protocols [11, 14] and at least quadratic in the number of jammed channels for randomized protocols [12, 29] if the adversary can jam almost all channels at a time). Recent work [10] also focuses on the wireless synchronization problem which requires devices to be activated at different times on a congested single-hop radio network to synchronize their round numbering while an adversary can disrupt a certain number of frequencies per round. Gilbert et al. [15] study robust information exchange in single-hop networks.

Our work is motivated by the work in [3] and [2]. In [3] it is shown that an adaptive jammer can dramatically reduce the throughput of the standard MAC protocol used in IEEE 802.11 with only limited energy cost on the adversary side. Awerbuch et al. [2] initiated the study of throughput-competitive MAC protocols under continuously running, adaptive jammers, but they only consider single-hop wireless networks. We go one step further by considering multi-hop networks where different nodes can have different channel states at a time, e.g., a transmission may be received only by a fraction of the nodes. It turns out that while the MAC protocol of [2] can be adopted to the multi-hop setting with a small modification, the proof techniques cannot. We are not aware of any other theoretical work on MAC protocols for multi-hop networks with provable performance against adaptive jamming.

### 1.3 Our Contributions

In this paper, we present a robust MAC protocol called Jade. Jade is a fairly simple protocol: it is based on a very small set of assumptions and rules and has a minimal storage overhead. In fact, in Jade every node just stores a constant number of parameters, among them a fixed parameter that should be chosen so that the following main theorem holds:

###### Theorem 1.1

When running Jade for time steps, Jade has a constant competitive throughput for any -bounded adversary and any UDG w.h.p. as long as and (a) the adversary is 1-uniform and the UDG is connected, or (b) there are at least nodes within the transmission range of every node.

Note that in practice, and are rather small so that our condition on is not too restrictive. Also, a conservative estimate on and would leave room for a superpolynomial change in and a polynomial change in over time.

On the other hand, we can also show the following result demonstrating that Theorem 1.1 essentially captures all the scenarios (within our notation) under which Jade can have a constant competitive throughput.

###### Theorem 1.2

If (a) the UDG is not connected, or (b) the adversary is allowed to be 2-uniform and there are nodes with nodes within their transmission range, then there are cases in which Jade is not constant competitive for any constant independent of .

Certainly, no MAC protocol can guarantee a constant competitive throughput if the UDG is not connected. However, it is still open whether there are simple MAC protocols that are constant competitive under non-uniform jamming strategies even if there are nodes within the transmission range of a node.

This section first gives a short motivation for our algorithmic approach and then presents the Jade protocol in detail.

### 2.1 Intuition

The intuition behind our MAC protocol is simple: in each round, each node tries to send a message with probability with for some small constant . Consider the unit disk around node consisting of ’s neighboring nodes as well as .111In this paper, disks (and later sectors) will refer both to 2-dimensional areas in the plane as well as to the set of nodes in the respective areas. The exact meaning will become clear in the specific context. Moreover, let and . Suppose that is sensing the channel. Let be the probability that the channel is idle at and let be the probability that exactly one node in is sending a message. It holds that and . Hence,

 q1≤∑v∈N(u)pv11−^p∏w∈N(u)(1−pw)=q0⋅p1−^p,  q1≥∑v∈N(u)pv∏w∈N(u)(1−pw)=q0⋅p.

Thus we have the following lemma, which has also been derived in [2] for the single-hop case.

###### Lemma 1

.

By Lemma 1, if a node observes that the number of rounds in which the channel is idle is essentially equal to the number of rounds in which exactly one message is sent, then is likely to be around 1 (if is a sufficiently small constant), which would be ideal. Otherwise, the nodes know that they need to adapt their probabilities. Thus, if we had sufficiently many cases in which an idle channel or exactly one message transmission is observed (which is the case if the adversary does not heavily jam the channel and is not too large), then one can adapt the probabilities just based on these two events and ignore all cases in which the wireless channel is blocked, either because the adversary is jamming it or because at least two messages interfere with each other (see also [19] for a similar conclusion). Unfortunately, can be very high for some reason, which requires a more sophisticated strategy for adjusting the access probabilities.

### 2.2 Protocol Description

In Jade, each node maintains a probability value , a threshold and a counter . The parameters in the protocol are fixed and the same for every node. may be set to any constant value so that , and should be small enough so that the condition in Theorem 1.1 is met.

Initially, every node sets , and . Afterwards, the Jade protocol works in synchronized rounds. In every round, each node decides with probability to send a message. If it decides not to send a message, it checks the following two conditions: If senses an idle channel, then . If successfully receives a message, then and . Afterwards, sets . If then it does the following: sets , and if there was no round among the past rounds in which sensed a successful message transmission or an idle channel, then and .

As we will see in the upcoming section, the concept of using a multiplicative-increase-multiplicative-decrease mechanism for and an additive-increase-additive-decrease mechanism for , as well as the slight modifications of the protocol in [2], marked in italic above, are crucial for Jade to work.

In contrast the description of Jade, its stochastic analysis is rather involved as it requires to shed light onto the complex interplay of the nodes all following their randomized protocol in a highly dependent manner. We first prove Theorem 1.1 (Sections 3.1 and 3.2) and then prove Theorem 1.2 (Section 3.3). In order to show the theorems, we will frequently use the following variant of the Chernoff bounds [2, 35].

###### Lemma 2

Consider any set of binary random variables . Suppose that there are values with for every set . Then it holds for and and any that

 P[X≥(1+δ)μ]≤(eδ(1+δ)1+δ)μ≤e−δ2μ2(1+δ/3).

If, on the other hand, it holds that for every set , then it holds for any that

 P[X≤(1−δ)μ]≤(e−δ(1−δ)1−δ)μ≤e−δ2μ/2.

Throughout the section we assume that is sufficiently small.

### 3.1 Proof of Theorem 1.1

We first look at a slightly weaker form of adversary. We say a round is open for a node if and at least one other node in its neighborhood are non-jammed (which implies that ’s neighborhood is non-empty). An adversary is weakly -bounded for some and if the adversary is -bounded and in addition to this, at least a constant fraction of the non-jammed rounds at each node are open in every time interval of size .

###### Theorem 3.1

When running Jade for rounds it holds w.h.p. that Jade is constant competitive for any weakly -bounded adversary.

###### Proof

First, we focus on a time frame consisting of subframes of size each, where is a multiple of and is a sufficiently large constant. The proof needs the following three lemmas. The first one is identical to Claim 2.5 in [2]. It is true because only successful message transmissions reduce .

###### Lemma 3

If in a time interval the number of rounds in which a node successfully receives a message is at most , then increases in at most rounds in .

The second lemma holds for arbitrary (not just weakly) -bounded adversaries and will be shown in Section 3.2.

###### Lemma 4

For every node , for at least a -fraction of the rounds in time frame , w.h.p., where the constant can be made arbitrarily small.

The third lemma just follows from some simple geometric argument.

###### Lemma 5

A disk of radius 2 can be cut into at most 20 regions so that the distance between any two points in a region is at most 1.

Consider some fixed node . Let be the set of all non-jammed open rounds at in time frame (which are a constant fraction of the non-jammed rounds at because we have a weakly -bounded adversary). Let be a constant satisfying Lemma 4 (i.e., ). Define to be the disk of radius 2 around (i.e., it has twice the radius of ). Cut into 20 regions satisfying Lemma 5, and let be any node in region (if such a node exists), where if . According to Lemma 4 it holds for each that at least a -fraction of the rounds in satisfy for any constant , w.h.p. Thus, at least a -fraction of the rounds in satisfy for every for any constant , w.h.p. As for all and has at least non-jammed rounds in , we get the following lemma, which also holds for arbitrary -bounded adversaries

###### Lemma 6

At least a -fraction of the rounds in satisfy and for all nodes for any constant , w.h.p.

Let us call these rounds good. Since the probability that senses the channel is at least and the probability that the channel at is idle for is equal to , senses an idle channel for at least many rounds in on expectation if is sufficiently small. This also holds w.h.p. when using the Chernoff bounds under the condition that at least rounds in are good (which also holds w.h.p.). Let be the number of times receives a message in . We distinguish between two cases.

Case 1: . Then Jade is constant competitive for and we are done.

Case 2: . Then we know from Lemma 3 that is decreased at most times in due to . In addition to this, is decreased at most times in due to a received message. On the other hand, is increased at least times in (if possible) due to an idle channel w.h.p. Also, we know from the Jade protocol that at the beginning of , . Hence, there must be at least rounds in w.h.p. at which . As there are at least good rounds in (w.h.p.), there are at least good rounds in w.h.p. in which . For these good rounds, has a constant probability to transmit a message and every node has a constant probability of receiving it, so successfully transmits messages to at least one of its non-jammed neighbors in (on expectation and also w.h.p.).

If we charge of each successfully transmitted message to the sender and to the receiver, then a constant competitive throughput can be identified for every node in both cases above, so Jade is constant competitive in .

It remains to show that Theorem 3.1 also holds for larger time intervals than . First, note that all the proofs are valid as long as for a constant , so we can increase and thereby also as long as this inequality holds. So w.l.o.g. we may assume that . In this case, , so our rule of increasing in Jade implies that at any time. This allows us to extend the competitive throughput result from a single to any sequence of polynomial in many time frames , which completes the proof of Theorem 3.1. ∎

Now, let us consider the two cases of Theorem 1.1. Recall that we allow here any -bounded adversary and not just a weakly bounded.

#### Case 1: the adversary is 1-uniform and the UDG is connected.

In this case, every node has a non-empty neighborhood and therefore all non-jammed rounds of the nodes are open. Hence, the conditions on a weakly -bounded adversary are satisfied. So Theorem 3.1 applies, which completes the proof of Theorem 1.1 a).

#### Case 2: |D(v)|≥2/ϵ for all v∈V.

Consider some fixed time interval with being a multiple of . For every node let be the number of non-jammed rounds at in and be the number of open rounds at in . Let be the set of rounds in with at most one non-jammed node. Suppose that . Then every node in must have more than of its non-jammed rounds in . As these non-jammed rounds must be serialized in to satisfy our requirement on , it holds that . Since this is impossible, it must hold that .

Thus, because . Let be the set of nodes with . That is, for each of these nodes, a constant fraction of the non-jammed time steps is open. Then , so .

Consider now a set of nodes so that and for every there are at most 6 nodes with ( is easy to construct in a greedy fashion for arbitrary UDGs and also known as a dominating set of constant density). Let . Since for every node , it follows that . Using that together with Theorem 3.1, which implies that Jade is constant competitive w.r.t. the nodes in , completes the proof of Theorem 1.1 b).

### 3.2 Proof of Lemma 4

In order to finish the proof of Theorem 1.1, it remains to prove Lemma 4. Consider any fixed node . We partition ’s unit disk into six sectors of equal angles from , . Note that all nodes within a sector have distances of at most 1 from each other, so they can directly communicate with each other (in , distances can be up to 2). We will first explore properties of an arbitrary node in one sector, then consider the implications for a whole sector, and finally bound the cumulative sending probability in the entire unit disk.

Recall the definition of a time frame, a subframe and in the proof of Theorem 3.1. Fix a sector in and consider some fixed time frame . Let us refer to the sum of the probabilities of the neighboring nodes of a given node by . The following lemma shows that will decrease dramatically if is high throughout a certain time interval.

###### Lemma 7

Consider a node in a unit disk . If during all rounds of a subframe of , then will be at most at the end of , w.h.p.

###### Proof

We say that a round is useful for node if from ’s perspective there is an idle channel or a successful transmission at that round (when ignoring the action of ); otherwise the round is called non-useful. Note that in a non-useful round, according to our protocol, will either decrease (if the threshold is exceeded) or remain the same. On the other hand, in a useful round, will increase (if senses an idle channel), decrease (if senses a successful transmission) or remain the same (if sends a message). Hence, can only increase during useful rounds of . Let be the set of useful rounds in for our node . We distinguish between two cases, depending on the cardinality . In the following, let denote the probability of at the beginning of (which is at most ). Suppose that for a sufficiently large constant . (This lower bound coincides with our definition of in the proof of Theorem 3.1.)

Case 1: Suppose that , that is, many rounds are blocked and can increase only rarely. As there are at least occasions in in which and , in at least of these occasions only saw blocked channels for consecutive rounds and therefore decides to increase and decrease . Hence, at the end of ,

 pv≤(1+γ)|U|−2clnn/γpv(0)≤(1+γ)−clnn/γpv(0)≤e−clnn=1/nc.

Case 2: Next, suppose that . We will show that many of these useful rounds will be successful such that decreases. Since throughout , it follows from the Chernoff bounds that w.h.p.  will sense the channel for at least a fraction of of the useful rounds w.h.p. Let this set of useful rounds be called . Consider any round . Let be the probability that there is an idle channel at round and be the probability that there is a successful transmission at . It holds that . From Lemma 1 we also know that . Since for all rounds in , it follows that for every round in . Thus, it follows from the Chernoff bounds that for at least of the rounds in , will sense a successful transmission w.h.p. Hence, at the end of it holds w.h.p. that

 pv≤(1+γ)−(1/3)⋅|U′|pv(0)≤(1+γ)−(1/3)⋅(2c/3)lnn/γpv(0)≤e−(2c/9)lnn=1/n2c/9.

Combining the two cases with results in the lemma. ∎

Given this property of the individual probabilities, we can derive a bound for the cumulative probability of an entire sector . In order to compute , we introduce three thresholds, a low one, , one in the middle, , and a high one, . The following three lemmas provide some important insights about these probabilities.

###### Lemma 8

For any subframe in and any initial value of in there is at least one round in with w.h.p.

###### Proof

We prove the lemma by contradiction. Suppose that throughout the entire interval , . Then it holds for every node that throughout . In this case, however, we know from Lemma 7, that will decrease to at most at the end of w.h.p. Hence, all nodes would decrease to at most at the end of w.h.p., which results in . This contradicts our assumption, so w.h.p. there must be a round in at which . ∎

###### Lemma 9

For any time interval in of size and any sector it holds that if at the beginning of , then throughout , w.m.p. Similarly, if at the beginning of , then throughout , w.m.p.

###### Proof

It suffices to prove the lemma for the case that initially as the other case is analogous. Consider some fixed round in . Let be the cumulative probability at the beginning of and be the cumulative probability at the end of . Moreover, let denote the cumulative probability of the nodes with no transmitting node in in round . Similarly, let denote the cumulative probability of the nodes with a single transmitting node in , and let be the cumulative probability of the nodes that experience a blocked round either because they are jammed or at least two nodes in are transmitting at . Certainly, . Our goal is to determine in this case. Let be the probability that all nodes in stay silent, be the probability that exactly one node in is transmitting, and be the probability that at least two nodes in are transmitting.

When ignoring the case that for a node at round , it holds:

 E[p′S] = q0(S)⋅[(1+γ)p(0)S+(1+γ)−1p(1)S+p(2)S] +q1(S)⋅[(1+γ)−1p(0)S+p(1)S+p(2)S] +q2(S)⋅[p(0)S+p(1)S+p(2)S]

This is certainly also an upper bound for if for a node because will never be increased (but possibly decreased) in this case. Now, consider the event that at least two nodes in transmit a message. If holds, then , so there is no change in the system. On the other hand, assume that does not hold. Let and be the probabilities and under the condition of . Then we distinguish between three cases.

Case 1: . Then

 E[p′S] ≤ q′0(S)⋅(1+γ)pS+q′1(S)⋅(1+γ)−1pS = ((1+γ)q′0(S)+(1+γ)−1q′1(S))pS.

From Lemma 1 we know that , so . If , then . Hence,

 E[p′S] ≤ ((1+γ)/6+(1+γ)−15/6)pS≤(1+γ)−1/2pS

since . On the other hand, in any case.

Case 2: . Then

 E[p′S] ≤ q′0(S)⋅(1+γ)−1pS+q′1(S)pS = (q′0(S)/(1+γ)+(1−q′0(S)))pS=(1−q′0(S)γ/(1+γ))pS.

Now, it holds that for all because from the Taylor series of and it follows that

 (1+γ)−x/2≥1−(xln(1+γ))/2≥1−(x(1−γ/2)γ)/2

and

 1−xγ/(1+γ)≤1−(x(1−γ/2)γ)/2

for all as is easy to check. Therefore, when defining , we get . On the other hand, .

Case 3: . Then for , and .

Combining the three cases and taking into account that , we obtain the following result.

###### Lemma 10

There is a (depending on , and ) so that

 E[p′S]≤(1+γ)−ϕpS% andp′S≤(1+γ)2ϕpS. (1)
###### Proof

Let , for the defined in Case 2, and . Furthermore, let , and . Define . Then we have

 E[p′S]≤(1+γ)−1/2p(0)S+(1+γ)−φ/2p(1)S+p(2)S=(1+γ)−ϕpS.

We need to show that for this , also . As , this is true if

 a2x0+b2x1+c2x2≤1((1/a)x0+(1/b)x1+(1/c)x2)2

or

 ((1/a)x0+(1/b)x1+(1/c)x2)2(a2x0+b2x1+c2x2)≤1 (2)

To prove this, we need two claims whose proofs are tedious but follow from standard math.

###### Claim

For any and any with ,

 (ax0+bx1+cx2)2≤(a2x0+b2x1+c2x2)
###### Claim

For any and any with ,

 ((1/a)x0+(1/b)x1+(1/c)x2)(ax0+bx1+cx2)≤1

Combining the claims, Equation 2 follows, which completes the proof. ∎

Hence, for any outcome of , and for some . If we define , then it holds that . For any time in , let be equal to at time and be defined as at time . Our calculations above imply that as long as , and .

Now, suppose that within subframe we reach a point when . Since we start with , there must be a time interval so that right before , , during we always have , and at the end of , . We want to bound the probability for this to happen.

Consider some fixed interval with the properties above, i.e., with right before and at the first round of , so initially, . We use martingale theory to bound the probability that in this case, the properties defined above for hold. Consider the rounds in to be numbered from 1 to , let and be defined as above, and let . It holds that

 E[q′t+1]=E[qt+1+t∑i=1φi]=E[qt+1]+t∑i=1φi≤qt−φt+t∑i=1φi=qt+t−1∑i=1φi=q′t.

Moreover, it follows from Inequality (1) that for any round , . Therefore, , which implies that . Hence, we can define a martingale with and that stochastically dominates . Recall that a random variable stochastically dominates a random variable if for any , . In that case, it is also straightforward to show that stochastically dominates , which we will need in the following. Let . We will make use of Azuma’s inequality to bound .

###### Fact 3.2 (Azuma Inequality)

Let be a martingale satisfying the property that for all . Then for any ,

 P[XT>X0+δ]≤e−δ2/(2∑Ti=1c2i).

Thus, for it holds in our case that

 P[XT>X0+δ]≤e−δ2/(2∑Ti=1φ2i).

This implies that

 P[q′T>q′0+δ]≤e−δ2/(2∑Ti=1φ2i),

for several reasons. First of all, stochastic dominance holds as long as , and whenever this is violated, we can stop the process as the requirements on would be violated, so we would not have to count that probability towards . Therefore,

 P[qT>q0+1/γ]≤e−δ2/(2∑Ti=1φ2